In the first scan, what should one suppose is meant by the way the tremolos for 2nd violins are notated (compare with the more normally notated equivalent tremolo for Flute 3)?

I think it means nothing special. The modern rules for notating tremolos were only established in the 20th century; in 19th century works, tremolos are often notated "wrongly" in many different ways. In this excerpt, only the first tremolo of the 3rd flute is correctly notated by modern standards. But I have to admit the 2nd violins' notation is one I've never seen before.

Being such a practical musician, Verdi was concerned about the players immediately understanding the length of each tremolo since they alternate between a dotted quarter and quarter in total duration. So he came up with something original. The flute didn't need such clarification since it only uses tremolos of whole and half measures. The following example from the same place in the opera might make this clearer:

erelievonen wrote:
In this excerpt, only the first tremolo of the 3rd flute is correctly notated by modern standards. But I have to admit the 2nd violins' notation is one I've never seen before.

I tried to recopy this in Sibelius and found I couldn’t get the beams the way Verdi wrote them with any setting. Then I realized what erelievonen meant: This half-measure duration requires two and the connected beams make them look like — Modern notation requires all 4 tremolo beams between stems, with none touching.

To me this is terribly cumbersome notation for a vanishingly tiny detail — a change in only the last of 24 notes of an extremely quiet measured tremolo — in a whole section of strings, no less. I wonder if this tiny passing tone is ever even heard. Here’s a different notational idea to maybe gain the same effect with much less clutter:

I think its absence would be heard, or at least sensed, even allowing for the fact that most members of the audience will be far more aware of the vocal line than of the orchestral detail. (Mind you, sopranos do seem altogether more willing/able than tenors to sing quietly when Verdi asks.)

I agree completely, David. The passing note is needed to connect the trills to the lower auxiliary into a smooth chain. And it is not clear that the trill ends on the principal note unless it is written in, thus my suggestion for improving MJCube's version.